Question

write explain how to use transformations to determine if figures are congruent or similar. remember images created from the same pre - image are always similar figures. practice 1. verify that the two figures are similar by describing a dilation that maps one figure onto the other. be sure to include the scale factor. a. scale factor written by hand b.

Explanation:

Step1: Recall dilation rule

A dilation with center \((0,0)\) and scale - factor \(k\) maps a point \((x,y)\) to \((kx,ky)\). To find the scale factor between two similar figures, we can compare the lengths of corresponding sides.

Step2: For part a

Let's consider the vertical side of the smaller trapezoid \(AB\) with endpoints \(A(2,2)\) and \(B(6,2)\) and the vertical side of the larger trapezoid \(CD\) with endpoints \(C(2,4)\) and \(D(6,4)\). The length of \(AB = 6 - 2=4\), and the length of \(CD = 6 - 2 = 4\). Now consider the non - vertical side. Let's take the slant side of the smaller trapezoid from \((6,2)\) to \((12,6)\) and the slant side of the larger trapezoid from \((6,4)\) to \((12,10)\).
We can see that if we dilate the smaller trapezoid with a scale factor of \(2\) centered at the origin \((0,0)\), a point \((x,y)\) on the smaller trapezoid will be mapped to \((2x,2y)\) on the larger trapezoid. For example, if we take a vertex \((2,2)\) of the smaller trapezoid, after dilation with \(k = 2\), it becomes \((2\times2,2\times2)=(4,4)\) (this is just an example of the rule application). The scale factor \(k = 2\).

Step3: For part b

Let's consider the side of the smaller rectangle \(EF\) with endpoints \(E(-6,4)\) and \(F(-4,2)\) and the side of the larger rectangle \(AD\) with endpoints \(A(-8,6)\) and \(D(-4,6)\).
First, find the length of \(EF\) using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(E(-6,4)\) and \(F(-4,2)\), \(d_{EF}=\sqrt{(-4+6)^2+(2 - 4)^2}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}\). For \(A(-8,6)\) and \(D(-4,6)\), \(d_{AD}=\sqrt{(-4 + 8)^2+(6 - 6)^2}=4\).
If we center the dilation at the origin \((0,0)\), and we want to map the smaller figure to the larger figure. Let's consider the ratio of the lengths of corresponding sides. If we take a point \((x,y)\) on the smaller figure and map it to \((kx,ky)\) on the larger figure. By comparing the side - lengths, we find that the scale factor \(k=\sqrt{2}\). For example, if we take a vertex \((-6,4)\) of the smaller figure, if \(k = \sqrt{2}\), the new point is \((-6\sqrt{2},4\sqrt{2})\). But if we center the dilation at \((- 2,2)\):
Let \((x,y)\) be a point on the smaller figure and \((x_0,y_0)=(-2,2)\) be the center of dilation. The formula for dilation is \((x',y')=(x_0+(x - x_0)k,y_0+(y - y_0)k)\).
For the smaller rectangle and the larger rectangle, if we center the dilation at \((-2,2)\), we can see that the scale factor \(k=\sqrt{2}\). For example, if we take point \(E(-6,4)\), \((x - x_0)=-6+2=-4\) and \((y - y_0)=4 - 2 = 2\). After dilation with \(k=\sqrt{2}\), \(x'=-2+( - 4)\sqrt{2}\) and \(y'=2+2\sqrt{2}\). Another way is to compare the distances of corresponding points from the center of dilation.
The scale factor for part a is \(2\) and for part b is \(\sqrt{2}\).

Answer:

a. The scale factor is \(2\). The dilation centered at the origin \((0,0)\) maps the smaller trapezoid to the larger trapezoid. Each coordinate of the vertices of the smaller trapezoid is multiplied by \(2\) to get the vertices of the larger trapezoid.
b. The scale factor is \(\sqrt{2}\). A dilation centered at \((-2,2)\) maps the smaller rectangle - like figure to the larger one. Each distance of a point on the smaller figure from the center \((-2,2)\) is multiplied by \(\sqrt{2}\) to get the corresponding point on the larger figure.